# cqplot: Chi Square Quantile-Quantile plots In heplots: Visualizing Hypothesis Tests in Multivariate Linear Models

## Description

A chi square quantile-quantile plots show the relationship between data-based values which should be distributed as χ^2 and corresponding quantiles from the χ^2 distribution. In multivariate analyses, this is often used both to assess multivariate normality and check for outliers, using the Mahalanobis squared distances (D^2) of observations from the centroid.

cqplot is a more general version of similar functions in other packages that produce chi square QQ plots. It allows for classical Mahalanobis squared distances as well as robust estimates based on the MVE and MCD; it provides an approximate confidence (concentration) envelope around the line of unit slope, a detrended version, where the reference line is horizontal, the ability to identify or label unusual points, and other graphical features.

The method for "mlm" objects applies this to the residuals from the model.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cqplot(x, ...) ## S3 method for class 'mlm' cqplot(x, ...) ## Default S3 method: cqplot(x, method = c("classical", "mcd", "mve"), detrend = FALSE, pch = 19, col = palette(), cex = par("cex"), ref.col = "red", ref.lwd = 2, conf = 0.95, env.col = "gray", env.lwd = 2, env.lty = 1, env.fill = TRUE, fill.alpha = 0.2, fill.color = trans.colors(ref.col, fill.alpha), labels = if (!is.null(rownames(x))) rownames(x) else 1:nrow(x), id.n, id.method = "y", id.cex = 1, id.col = palette(), xlab, ylab, main, what=deparse(substitute(x)), ylim, ...) 

## Arguments

 x either a numeric data frame or matrix for the default method, or an object of class "mlm" representing a multivariate linear model. In the latter case, residuals from the model are plotted. ... Other arguments passed to methods method estimation method used for center and covariance, one of: "classical" (product-moment), "mcd" (minimum covariance determinant), or "mve" (minimum volume ellipsoid). detrend logical; if FALSE, the plot shows values of D^2 vs. χ^2. if TRUE, the ordinate shows values of D^2 - χ^2 pch plot symbol for points Can be a vector of length equal to the number of rows in x. col color for points; the default is the first entry in the current color palette (see palette and par. cex character symbol size for points. Can be a vector of length equal to the number of rows in x. ref.col Color for the reference line ref.lwd Line width for the reference line conf confidence coverage for the approximate confidence envelope env.col line color for the boundary of the confidence envelope env.lwd line width for the confidence envelope env.lty line type for the confidence envelope env.fill logical; should the confidence envelope be filled? fill.alpha transparency value for fill.color fill.color color used to fill the confidence envelope labels vector of text strings to be used to identify points, defaults to rownames(x) or observation numbers if rownames(x) is NULL id.n number of points labeled. If id.n=0, the default, no point identification occurs. id.method point identification method. The default id.method="y" will identify the id.n points with the largest value of abs(y-mean(y)). See showLabels for other options. id.cex size of text for point labels id.col color for point labels xlab label for horizontal (theoretical quantiles) axis ylab label for vertical (empirical quantiles) axis main plot title what the name of the object plotted; used in the construction of main when that is not specified. ylim limits for vertical axis. If not specified, the range of the confidence envelope is used.

## Details

The calculation of the confidence envelope follows that used in the SAS program, http://www.datavis.ca/sasmac/cqplot.html which comes from Chambers et al. (1983), Section 6.8.

The essential formula is

SE ( z_{(i)} ) = \frac{\hat{δ}}{g ( q_i )) \times √{ frac{ p_i (1-p_i} }{n}}

where z_{(i)} is the i-th order value of D^2, \hat{δ} is an estimate of the slope of the reference line obtained from the corresponding quartiles and g(q_i) is the density of the chi square distribution at the quantile q_i.

Note that this confidence envelope applies only to the D^2 computed using the classical estimates of location and scatter. The car::qqPlot() function provides for simulated envelopes, but only for a univariate measure. Oldford (2016) provides a general theory and methods for QQ plots.

## Value

Returns invisibly the vector of squared Mahalanobis distances corresponding to the rows of x or the residuals of the model.

Michael Friendly

## References

J. Chambers, W. S. Cleveland, B. Kleiner, P. A. Tukey (1983). Graphical methods for data analysis, Wadsworth.

R. W. Oldford (2016), "Self calibrating quantile-quantile plots", The American Statistician, 70, 74-90.

Mahalanobis for calculation of Mahalanobis squared distance;
qqplot; qqPlot can give a similar result for Mahalanobis squared distances of data or residuals; qqtest has many features for all types of QQ plots.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 cqplot(iris[, 1:4]) iris.mod <- lm(as.matrix(iris[,1:4]) ~ Species, data=iris) cqplot(iris.mod, id.n=3) # compare with car::qqPlot car::qqPlot(Mahalanobis(iris[, 1:4]), dist="chisq", df=4) # Adopted data Adopted.mod <- lm(cbind(Age2IQ, Age4IQ, Age8IQ, Age13IQ) ~ AMED + BMIQ, data=Adopted) cqplot(Adopted.mod, id.n=3) cqplot(Adopted.mod, id.n=3, method="mve") # Sake data Sake.mod <- lm(cbind(taste, smell) ~ ., data=Sake) cqplot(Sake.mod) cqplot(Sake.mod, method="mve", id.n=2) # SocialCog data -- one extreme outlier data(SocialCog) SC.mlm <- lm(cbind(MgeEmotions,ToM, ExtBias, PersBias) ~ Dx, data=SocialCog) cqplot(SC.mlm, id.n=1) # data frame example: stackloss data data(stackloss) cqplot(stackloss[, 1:3], id.n=4) # very strange cqplot(stackloss[, 1:3], id.n=4, detrend=TRUE) cqplot(stackloss[, 1:3], id.n=4, method="mve") cqplot(stackloss[, 1:3], id.n=4, method="mcd")